| L(s) = 1 | + (1.28 + 0.587i)2-s + (−0.576 + 1.96i)3-s + (1.30 + 1.51i)4-s + (3.46 + 3.60i)5-s + (−1.89 + 2.18i)6-s + (0.457 − 3.18i)7-s + (0.796 + 2.71i)8-s + (4.04 + 2.59i)9-s + (2.34 + 6.67i)10-s + (−18.0 + 8.26i)11-s + (−3.72 + 1.70i)12-s + (14.1 − 2.03i)13-s + (2.45 − 3.82i)14-s + (−9.07 + 4.73i)15-s + (−0.569 + 3.95i)16-s + (8.56 − 9.88i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.293i)2-s + (−0.192 + 0.654i)3-s + (0.327 + 0.377i)4-s + (0.693 + 0.720i)5-s + (−0.316 + 0.364i)6-s + (0.0653 − 0.454i)7-s + (0.0996 + 0.339i)8-s + (0.449 + 0.288i)9-s + (0.234 + 0.667i)10-s + (−1.64 + 0.751i)11-s + (−0.310 + 0.141i)12-s + (1.08 − 0.156i)13-s + (0.175 − 0.273i)14-s + (−0.605 + 0.315i)15-s + (−0.0355 + 0.247i)16-s + (0.503 − 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.47482 + 1.76448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47482 + 1.76448i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.587i)T \) |
| 5 | \( 1 + (-3.46 - 3.60i)T \) |
| 23 | \( 1 + (22.0 + 6.55i)T \) |
| good | 3 | \( 1 + (0.576 - 1.96i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.457 + 3.18i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (18.0 - 8.26i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-14.1 + 2.03i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-8.56 + 9.88i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (4.81 - 4.16i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (15.7 - 18.1i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-50.2 + 14.7i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (23.8 + 15.2i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-30.5 + 19.6i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-77.6 - 22.7i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 + 61.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-10.8 + 75.6i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (7.33 + 51.0i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (14.8 + 50.6i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (28.2 - 61.8i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-43.9 + 96.1i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-16.7 + 14.5i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-15.8 + 2.27i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (31.8 + 20.4i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (41.3 - 140. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (52.8 - 33.9i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.46250424688100098298811905930, −10.96859698932776257480000613273, −10.47187732032348051407884655932, −9.703627005652324775146475694769, −7.999532230253522221124993874080, −7.14634126189732035810632240866, −5.88412204329755246468896521561, −4.98648944419733870546855902681, −3.77481326779789921903652281757, −2.30842709289330278050616991883,
1.14538667418791576437105120923, 2.56300378333640083150638896793, 4.27289793406121240235638819742, 5.75638756584391045643437074623, 6.05346066830721674742645064677, 7.72410900271714985426950829206, 8.689422394771344766875522309819, 9.978893942671948334084171181268, 10.84748258321415522386912458619, 12.06316928349541003360348742765
